Analyze The Polynomial Function F ( X ) = X ( 4 − X ) ( 7 − X F(x) = X(4-x)(7-x F ( X ) = X ( 4 − X ) ( 7 − X ] Using Parts (a) Through (h) Below.(a) Determine The End Behavior Of The Graph Of The Function.The Graph Of F F F Behaves Like Y = □ Y = \square Y = □ For Large Values Of ∣ X ∣ |x| ∣ X ∣ .

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Introduction

In this analysis, we will be examining the polynomial function $f(x) = x(4-x)(7-x)$ using various methods to gain a deeper understanding of its behavior. The function is a product of three linear factors, which makes it a cubic polynomial. We will be using parts (a) through (h) to analyze the function, starting with the end behavior of the graph.

(a) Determine the End Behavior of the Graph of the Function

The end behavior of a function refers to the behavior of the function as $x$ approaches positive or negative infinity. To determine the end behavior of the graph of $f(x) = x(4-x)(7-x)$, we need to examine the leading term of the function.

The leading term of the function is the term with the highest degree, which in this case is $x$. As $x$ approaches positive or negative infinity, the term $x$ dominates the function, and the function behaves like $y = x$.

However, we also need to consider the other two factors, $(4-x)$ and $(7-x)$. As $x$ approaches positive infinity, both of these factors approach zero, and the function behaves like $y = x$. As $x$ approaches negative infinity, the factor $(4-x)$ approaches zero, and the function behaves like $y = -x(7-x)$.

Therefore, the graph of $f$ behaves like $y = x$ for large values of $|x|$.

(b) Find the Zeros of the Function

The zeros of a function are the values of $x$ for which the function is equal to zero. To find the zeros of the function $f(x) = x(4-x)(7-x)$, we need to set the function equal to zero and solve for $x$.

Setting the function equal to zero, we get:

$$x(4-x)(7-x) = 0$$

This equation is true when any of the factors are equal to zero. Therefore, the zeros of the function are:

$$x = 0, 4, 7$$

These are the values of $x$ for which the function is equal to zero.

(c) Find the x-Intercepts of the Graph

The x-intercepts of a graph are the points where the graph intersects the x-axis. To find the x-intercepts of the graph of $f(x) = x(4-x)(7-x)$, we need to find the values of $x$ for which the function is equal to zero.

As we found in part (b), the zeros of the function are $x = 0, 4, 7$. Therefore, the x-intercepts of the graph are:

$$(0, 0), (4, 0), (7, 0)$$

These are the points where the graph intersects the x-axis.

(d) Find the y-Intercept of the Graph

The y-intercept of a graph is the point where the graph intersects the y-axis. To find the y-intercept of the graph of $f(x) = x(4-x)(7-x)$, we need to find the value of $y$ when $x = 0$.

Substituting $x = 0$ into the function, we get:

$$f(0) = 0(4-0)(7-0) = 0$$

Therefore, the y-intercept of the graph is:

$$(0, 0)$$

This is the point where the graph intersects the y-axis.

(e) Determine the Domain of the Function

The domain of a function is the set of all possible input values for which the function is defined. To determine the domain of the function $f(x) = x(4-x)(7-x)$, we need to consider the restrictions on the input values.

The function is defined for all real numbers except for the values of $x$ that make the function equal to zero. As we found in part (b), the zeros of the function are $x = 0, 4, 7$. Therefore, the domain of the function is:

$$x \in (-\infty, 0) \cup (0, 4) \cup (4, 7) \cup (7, \infty)$$

This is the set of all possible input values for which the function is defined.

(f) Determine the Range of the Function

The range of a function is the set of all possible output values for which the function is defined. To determine the range of the function $f(x) = x(4-x)(7-x)$, we need to consider the behavior of the function as $x$ approaches positive or negative infinity.

As we found in part (a), the graph of $f$ behaves like $y = x$ for large values of $|x|$. Therefore, the range of the function is:

$$y \in (-\infty, \infty)$$

This is the set of all possible output values for which the function is defined.

(g) Determine the End Behavior of the Function

The end behavior of a function refers to the behavior of the function as $x$ approaches positive or negative infinity. To determine the end behavior of the function $f(x) = x(4-x)(7-x)$, we need to examine the leading term of the function.

The leading term of the function is the term with the highest degree, which in this case is $x$. As $x$ approaches positive or negative infinity, the term $x$ dominates the function, and the function behaves like $y = x$.

However, we also need to consider the other two factors, $(4-x)$ and $(7-x)$. As $x$ approaches positive infinity, both of these factors approach zero, and the function behaves like $y = x$. As $x$ approaches negative infinity, the factor $(4-x)$ approaches zero, and the function behaves like $y = -x(7-x)$.

Therefore, the graph of $f$ behaves like $y = x$ for large values of $|x|$.

(h) Determine the Local Maxima and Minima of the Function

The local maxima and minima of a function are the points where the function has a maximum or minimum value. To determine the local maxima and minima of the function $f(x) = x(4-x)(7-x)$, we need to examine the behavior of the function as $x$ approaches the zeros of the function.

As we found in part (b), the zeros of the function are $x = 0, 4, 7$. To determine the local maxima and minima of the function, we need to examine the behavior of the function as $x$ approaches these values.

As $x$ approaches $0$ from the left, the function approaches $-\infty$. As $x$ approaches $0$ from the right, the function approaches $+\infty$. Therefore, the function has a local maximum at $x = 0$.

As $x$ approaches $4$ from the left, the function approaches $-\infty$. As $x$ approaches $4$ from the right, the function approaches $+\infty$. Therefore, the function has a local maximum at $x = 4$.

As $x$ approaches $7$ from the left, the function approaches $-\infty$. As $x$ approaches $7$ from the right, the function approaches $+\infty$. Therefore, the function has a local maximum at $x = 7$.

Therefore, the local maxima and minima of the function are:

$$(0, \infty), (4, \infty), (7, \infty)$$

These are the points where the function has a maximum or minimum value.

Conclusion

In this analysis, we have examined the polynomial function $f(x) = x(4-x)(7-x)$ using various methods. We have determined the end behavior of the graph, found the zeros and x-intercepts of the function, determined the domain and range of the function, and determined the local maxima and minima of the function.

The function is a cubic polynomial with a leading term of $x$. As $x$ approaches positive or negative infinity, the function behaves like $y = x$. The function has zeros at $x = 0, 4, 7$, and the domain of the function is $x \in (-\infty, 0) \cup (0, 4) \cup (4, 7) \cup (7, \infty)$. The range of the function is $y \in (-\infty, \infty)$, and the local maxima and minima of the function are $(0, \infty), (4, \infty), (7, \infty)$.

This analysis has provided a deeper understanding of the behavior of the polynomial function $f(x) = x(4-x)(7-x)$.

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Introduction

In our previous article, we analyzed the polynomial function $f(x) = x(4-x)(7-x)$ using various methods. We determined the end behavior of the graph, found the zeros and x-intercepts of the function, determined the domain and range of the function, and determined the local maxima and minima of the function.

In this article, we will answer some frequently asked questions about the polynomial function $f(x) = x(4-x)(7-x)$.

Q: What is the degree of the polynomial function $f(x) = x(4-x)(7-x)$?

A: The degree of the polynomial function $f(x) = x(4-x)(7-x)$ is 3, since it is a product of three linear factors.

Q: What are the zeros of the polynomial function $f(x) = x(4-x)(7-x)$?

A: The zeros of the polynomial function $f(x) = x(4-x)(7-x)$ are $x = 0, 4, 7$, since these are the values of $x$ for which the function is equal to zero.

Q: What is the domain of the polynomial function $f(x) = x(4-x)(7-x)$?

A: The domain of the polynomial function $f(x) = x(4-x)(7-x)$ is $x \in (-\infty, 0) \cup (0, 4) \cup (4, 7) \cup (7, \infty)$, since the function is defined for all real numbers except for the values of $x$ that make the function equal to zero.

Q: What is the range of the polynomial function $f(x) = x(4-x)(7-x)$?

A: The range of the polynomial function $f(x) = x(4-x)(7-x)$ is $y \in (-\infty, \infty)$, since the function can take on any real value.

Q: What are the local maxima and minima of the polynomial function $f(x) = x(4-x)(7-x)$?

A: The local maxima and minima of the polynomial function $f(x) = x(4-x)(7-x)$ are $(0, \infty), (4, \infty), (7, \infty)$, since these are the points where the function has a maximum or minimum value.

Q: How can I graph the polynomial function $f(x) = x(4-x)(7-x)$?

A: You can graph the polynomial function $f(x) = x(4-x)(7-x)$ by using a graphing calculator or by plotting the function on a coordinate plane. You can also use software such as Mathematica or Maple to graph the function.

Q: Can I use the polynomial function $f(x) = x(4-x)(7-x)$ to model real-world phenomena?

A: Yes, you can use the polynomial function $f(x) = x(4-x)(7-x)$ to model real-world phenomena such as population growth, chemical reactions, and electrical circuits.

Q: How can I use the polynomial function $f(x) = x(4-x)(7-x)$ to solve problems?

A: You can use the polynomial function $f(x) = x(4-x)(7-x)$ to solve problems such as finding the maximum or minimum value of a function, determining the domain and range of a function, and graphing a function.

Conclusion

In this article, we have answered some frequently asked questions about the polynomial function $f(x) = x(4-x)(7-x)$. We have discussed the degree of the function, the zeros of the function, the domain and range of the function, and the local maxima and minima of the function. We have also provided information on how to graph the function and how to use the function to model real-world phenomena.

We hope that this article has been helpful in providing a deeper understanding of the polynomial function $f(x) = x(4-x)(7-x)$.